Claim: The fractional part of $\frac{2z}{i}$, where $z$ and $i$ are finite is no more than $\frac{i-2}{i}$ when $i$ is even and no more than $\frac{i-1}{i}$ when $i$ is odd. Here $z,i$ are non-negative integers. To make the problem non-trivial, assume $i \ge 3$.
Example: If $i=4$, then $2z$ can be $0,2,4,6,...12...$. The fractional parts for these numbers then are $0,2/4$. The largest of this is 2/4 which is $\frac{i-2}{i}$.
Second example: If $i=7$, then $2z$ can be $0,2,4,6,...42...$. The fractional parts for these numbers then are $0,2/7, 4/6,6/7,3/7,5/7$. The largest of this is 6/7 which is $\frac{i-1}{i}$.
I cannot write this simple result in the form of a rigorous proof. How should I?
UPDATE: I have provided an answer myself. Corrections are welcome.